“Shoshichi Kobayashi as Adviser,” Prof. Emeritus Gary Jensen

For me Shoshichi Kobayashi was the ideal thesis adviser and mentor. He played such a vital role at so many stages that I think it is accurate to say that I would never have become a professor of mathematics without his help and guidance. Here is the story.

After completing my qualifying exams in October, I spent the rest of the 1964-65 academic year doing more course work and wondering what I should do next. I had very little idea of what “writing a thesis” was all about. At the end of the spring semester, when a graduating friend asked me what I was doing, he responded with the suggestion that I pursue a different area of mathematics. He urged me to find a thesis adviser promptly. This was a daunting step for me and many other graduate students at that time. Fortunately, I had taken a representation theory course from Professor Kobayashi. He had impressed me as a professor whom I might actually talk to. I felt no anxiety in asking him to be my adviser. He didn’t really give me an answer at that time, but he suggested that I spend the summer reading his new book written with Nomizu, Foundations of Differential Geometry vol. I. Thus I began my lifelong love of differential geometry.

Not only had Professor Kobayashi suggested that I read the book, but he also suggested that I come to his office once a week to talk to him about what I was reading. Somehow I thought he really meant that I should come to talk when I had a question or some brilliant comment to make. In less than two weeks he emphatically set me straight on that issue. So I began intense reading, from page one to the end, combined with regular talking. Our talks were never very long. I still don’t know if that was his style, or if he recognized early on that I didn’t absorb things through detailed discussion very well. Most importantly, he knew when to prod me along, either gently or forcefully, depending on the situation.

At the beginning of the fall semester he agreed to be my adviser. I had no idea of what I wanted to work on, not even of what it meant to work on something. He handed me a list of problems he and James Eells had edited for the proceedings of a recent conference in Kyoto, Japan, with an indication of two or three problems that might interest me. A week later we agreed that I look at the problem proposed by Eells and Sampson: “Does any simply connected, compact Riemannian space of nonnegative curvature admit a Ricci parallel metric?”. During that academic year I read papers and got nowhere. When I finally told Professor Kobayashi that I felt I was making no progress, in fact, that I really had no idea of how to begin trying to solve this problem, his reply was simple and profoundly helpful. “Why don’t you just look at the dimension four homogeneous case first”, he said. It is embarrassing to remember that for a year this idea had not occurred to me. Along with this advice he suggested a paper by S. Ishihara on four-dimensional homogeneous spaces. At last, a paper I not only understood in just a couple of weeks, but which I saw how to apply to my problem.

Graduate school at Berkeley in the mid-sixties was wonderful. In our meeting at the beginning of the fall 1967 semester, Professor Kobayashi quietly mentioned that this would be my last year. “But I don’t have enough results for a thesis”, I reminded him. Yes, he agreed, but he stated again that this would be my last year. I got the message and it was a powerful motivator. In our weekly meetings I started presenting partial results, bits and pieces that at first seemed to hit a wall, but soon began yielding to the assault. By the end of December I had found all homogeneous Einstein spaces of dimension four.

Meanwhile, Professor Kobayashi had raised the issue of a job for next year. Strangely enough, I was very vague on this point, no doubt due to my subconscious desire to remain a graduate student for the rest of my life. He introduced me to some mathematicians from a university in the east and I told them I would like to join their department. Weeks passed with no communication from that department. I didn’t give it much thought, but one day Professor Kobayashi asked me, with some anxiety, whether I had heard anything. Hearing the answer, and hearing that I had not applied for anything else, he took me by the arm and escorted me to the library where notebooks contained lists of available jobs. Together we picked out a few. He told me to write a letter of application to each and to find two more people to write letters of recommendation for me. It still frightens me to think what might have become of me if he had not intervened so effectively at that time to make sure that I had applied for jobs. In early March I interviewed at Carnegie-Mellon and accepted their offer of a tenure track position. The Chair told me it was common practice for a candidate to ask about the salary before accepting the offer.

In June my family and I headed out for Pittsburgh with a copy of the manuscript of Foundations of Differential Geometry vol. II in the trunk of the car. After reading it I struggled to find a new research problem. During the year I corresponded fairly regularly with Professor Kobayashi. After a few months I told him that I felt isolated being the only person in the department interested in differential geometry. I asked him if a post-doctoral fellowship somewhere might be possible. I applied for a couple of these, without success In early March a call came from Washington University in St. Louis asking me if I would be interested in coming there to interview for a one year post-doc position connected to a large ONR funded special year in symmetric spaces. Professor Kobayashi had suggested my name to them for this position.

At the end of my post-doc year, Washington University offered me a tenure track position. I couldn’t resist this department with its strong, active geometry group, so we stayed in St. Louis.

Professor Kobayashi’s mentoring didn’t stop there. He was instrumental in arranging a visiting research position for me at Berkeley during the summer of 1971. Again he offered me a very simple bit of advice when I complained of being stuck in the process of writing up a paper. “Don’t try to explain all cases at once”, he said. In another conversation he directed my attention to his 1963 Tohoku Math. J. paper “Topology of positively pinched Kaehler manifolds”, which formed the basis of my best known paper of the 1970’s, “Einstein metrics on principal fiber bundles”, a paper that probably tipped the tenure decision in my favor. And so his quiet help continued over the years. Now I am a professor emeritus at Washington University.

What is clear to me about this story is that Professor Kobayashi kept my career, even my life, on track at several crucial points. I fear now that I never adequately communicated my awareness of this to him, even though we saw each other every few years when I would visit Berkeley. It is reasonable to assume that he understood my gratitude for all he had done for me, as his actions clearly indicated a good knowledge of my personality.